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A.11(B)·Algebra II·Exponential & Logarithmic FunctionsSupporting

Exponential Parameter Changes

Predict the effects of parameter changes on graphs of exponential functions.

Students distinguish output multiplication from output addition in exponential rules and use point mappings to classify vertical stretches and shrinks.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutorAll Algebra II released practice50 minutes

Learn the concept

Multiply outputs

If g(x)=af(x)g(x)=a f(x), each point (x,y)(x,y) maps to (x,ay)(x,ay). For a>1a>1 this is a vertical stretch; for 0<a<10<a<1 it is a vertical shrink. A negative multiplier also reflects the graph across the x-axis.

Add outputs

If h(x)=f(x)+kh(x)=f(x)+k, each point maps to (x,y+k)(x,y+k). This is a vertical translation, not a scaling. For f(x)=bxf(x)=b^x, a nonzero vertical multiplier keeps the horizontal asymptote y=0y=0, whereas adding k changes it to y=ky=k.

Compare point patterns

Use two or more inputs to distinguish multiplication from addition. Scaling has a constant ratio of corresponding outputs; translation has a constant difference. For exponentials, the new y-intercept under scaling is a because ab0=aa b^0=a.

Worked example

How does g(x)=6(3x)g(x)=6(3^x) compare with f(x)=3xf(x)=3^x? Contrast it with h(x)=3x+6h(x)=3^x+6.

  1. For f, the points at inputs 0 and 1 are (0,1)(0,1) and (1,3)(1,3).
  2. Multiply outputs by 6 for g: (0,6)(0,6) and (1,18)(1,18).
  3. Add 6 instead for h: (0,7)(0,7) and (1,9)(1,9).
  4. The output ratios for g are 6; h has output differences of 6.

g is a vertical stretch by factor 6, while h is shifted up 6 units.

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 1 Writing Exponential Functions from Real-world Situations

Five Minute Math · Aaron Daffern, Dallas ISD

Review the role of an exponential model’s outside coefficient as its initial value before comparing vertical scalings.

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TEKSForge practice · unscored

  1. 1. Describe g(x)=25(7x)g(x)=\frac25(7^x) relative to f(x)=7xf(x)=7^x.

    Show answer and reasoning

    A vertical shrink by factor 25\frac25.

    Each output becomes two-fifths of its previous value while the input stays fixed.

  2. 2. For f(x)=2xf(x)=2^x, compare g(x)=3f(x)g(x)=3f(x) and h(x)=f(x)+3h(x)=f(x)+3 at x=2x=2.

    Show answer and reasoning

    g(2)=12g(2)=12 and h(2)=7h(2)=7.

    f(2)=4f(2)=4. Multiplying gives 3(4)=123(4)=12, while adding gives 4+3=74+3=7, confirming different transformations.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 46 · 1 point

Which type of transformation can be used to obtain the graph of g(x)=4(2x)g(x) = 4(2^x) from the graph of f(x)=2xf(x) = 2^x?

  1. Vertical shrink
  2. Vertical shift down
  3. Vertical shift up
  4. Vertical stretch
Show STAAR answer and explanation

Vertical stretch

Multiplying every output by 4 stretches the graph vertically by a factor of 4: (0,1)(0,1) moves to (0,4)(0,4) and (1,2)(1,2) moves to (1,8)(1,8).

Vertical shrink: A vertical shrink multiplies outputs by a factor between 0 and 1.

Vertical shift down: A shift down subtracts a constant from every output.

Vertical shift up: A shift up would add 4, giving 2x+42^x+4, which moves (1,2)(1,2) to (1,6)(1,6), not (1,8)(1,8).

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Learning objectives

By the end of the lesson, students can:

  • Describe a vertical stretch or shrink of an exponential graph.
  • Map points under output multiplication.
  • Distinguish a multiplier from a vertical translation.

Instructional sequence

  1. 1

    Engage

    5 min

    Compare doubling an output with adding two to it.

  2. 2

    Direct instruction

    10 min

    Model the point map for an outside multiplier.

  3. 3

    Guided practice

    13 min

    Build tables for an exponential and its scaled versions.

  4. 4

    Compare and verify

    15 min

    Contrast scaling with a vertical shift using asymptotes and points.

  5. 5

    Closure

    7 min

    Justify the transformation on the exit ticket.

Misconceptions to address

“An outside factor shifts the graph up.”

Teacher move: Multiplication changes output ratios; addition changes every output by the same amount.

“Any positive factor makes a stretch.”

Teacher move: Factors strictly between zero and one shrink vertically; factors greater than one stretch.